Probability of specific XOR value (hints only please)

Question

I have a set with $5$ unique elements chosen randomly from the set of six-bit binary numbers, excluding $000000$. I take the power set of this set (i.e. the set of all subsets) without the empty set, and for each subset in the power set, I take the bitwise XOR of all of its members. What is the probability that one of the subsets XORs to a given value (say, $100010$)?

Please give me just a hint and not the whole solution. I feel that the solution includes something to do with representing the XOR operation as addition modulo $2$, but I don't exactly see how this works; am I going in the right direction? Thanks for your attention!

Answer 1

What I would do is:

• First, show that given a number $a$, for all number $c\neq a$, there exists a number $b$ such that $a \oplus b =c$.
• deduce the probability to obtain a particular number from two unique elements chosen randomly.
• iterate this to get the probability for 5 elements.

You may need to discuss at some point whether it is possible or not to have $a_1\oplus a_2\dots \oplus a_n= a_1$ and/or $a_1\oplus a_2\dots \oplus a_n= 0$.

There is surely a solution that use the addition modulo 2, but this one came to my mind first.

I hope my hints are (right and) clear enough.

EDIT By 'iterate' I mean look at the possible values that you can obtain from $a_1\oplus a_2\oplus a_3$: it cannot be $a_1$ (otherwise $a_3=a_2$) etc.

Then you continue for $a_1\oplus a_2\oplus a_3\oplus a_4$ with the same reasoning and again for $a_1\oplus a_2\oplus a_3\oplus a_4\oplus a_5$